In
this part of the problem students investigate vertical translations and
discover that the graph of the function y = f(x) + b
is achieved by translating the graph of the function y = f(x)
by vector (0, b) for any function f(x). They start
working with the graph of the function and then
generalize for an arbitrary f(x). Here are the steps of
construction and questions.

1.Open a new file. If axes do not
appear in the blank document, create them by clicking the Toggle grid and axes
icon on the top toolbar.

2.Select Function from the Draw
toolbox. Type: sqrt(X) for the function.

3.Draw a second function and
enter: sqrt(X)+b for the function. Select the curve, right-click and
select Properties from the context menu to change the line color to
Blue.

Q1.
As you drag the graph of function up and
down along the y-axis, what do you observe about the sign of parameter b
with respect to the position of this graph relative to the graph ?

A: It is expected that students will make the
following conjecture: b > 0 when the graph of is
above the graph of ; and b
< 0 when the graph of is below
the graph of .

In
order to see that, students can observe the value of b in the Variables
toolbox while dragging the graph of the function.

Q2.
What is the geometric
transformation from the graph of to.

A.
It is expected that students will
bring up translation by a vertical vector.

In
order to verify this with the software, follow these steps.

1.Select Vector from the Draw
toolbox. Construct a vector AB anywhere on the plane. Click the selection
arrow.

2.Click the graph of the function. Select Translation
from the Construct toolbox, select vector AB and the translation of
the graph will appear on the screen.

Q3.
What are the coordinates of the
vector of translation that transforms graph of to the graph
of ? Make a
conjecture and verify it with the help of the software.

A.
The teacher may help students come
up with a conjecture by pointing to the characteristic points of the graph,
in this case point (0, 0) which is the vertex of the semi-parabola. This
hint should help them determine the coordinates of the vector of translation.
Students will also use the software to help find the answer and verify their
conjecture.

1.Select vector AB, choose Coefficients
from the Constrain toolbox. The coefficients of
the vector will appear in the edit box. Accept the default values by pressing
Enter.

2.By moving the endpoint of the
vector B students can overlap the translated image with the graph of the
function and
observe the values of u0, v0, and b
in the Variables toolbox to answer the question and to verify their
conjecture.

The
expected answer: u0 = 0, v0 = b.

3.In order to verify this,
students can double click the coefficients and substitute these values. The
translated image will overlap with the graph of the function . Drag
point B to confirm that.

Q4. What vector will translate the graph of function back to
the graph of function ? Make a
conjecture and verify with the software.

A. The vector of translation is (0, -b).

Q5.
Generalize these results for an
arbitrary function f(x) and confirm your conclusion with the
help of the software.

A. Students should conclude that the graph of function
y = f(x) + b is derived from graph of function y
= f(x) by translation along the y-axis by the vector (0, b).
They should also conclude that the graph of function y = f(x)
is derived from graph of function y = f(x) + b by
translation along the y-axis by the vector (0, -b). They can verify this
by editing the equation of the function:

1.Double click the equation of the
function and change it to: f(x). The graph of the arbitrary function
will appear on the screen.

2.Click the image of the translated
function and choose Implicit Equation from the Calculate toolbox.
The equation y = b + f(x) will appear on the screen.

Similarly,
students can verify inverse translation with the help of the software.

In
this part of the problem students investigate horizontal translations and
discover that the graph of function y = f(x – a) is
always derived by translation from the graph of the function y = f(x)
by the vector (a, 0) for any function f(x). They start
working with the graph of the function and then
generalize for an arbitrary f(x). Here are the steps of
construction and questions.

1.Open a new file. If axes do not
appear in the blank document, create them by clicking the Toggle grid and axes
icon on the top toolbar.

2.Select Function from the Draw
toolbox. Type: sqrt(X) for the function.

3.Select Function from the Draw
toolbox. Type: sqrt(X - a) for the function. Select the function,
right-click and select Properties from the context menu to change the line
color to Blue.

Q1.
As you drag the graph of the function
left and
right along the x-axis, what do you observe about the sign of parameter a
with respect to the position of this graph relative to the graph ?

A: It is expected that students will make the
following conjecture: if a > 0, the graph of is
shifted to the right of the graph of ; if a
< 0, the graph of is
shifted to the left of the graph of .

In
order to see this, students can observe the value of a in the Variables
toolbox while dragging the graph of the function.

Q2.
What is the geometric
transformation from the graph of to.

A.
It is expected that students will
bring up translation by a horizontal vector.

In
order to verify this with the software, follow these steps:

1.Select Vector from the Draw
toolbox. Construct a vector AB anywhere on the plane. Click the selection
arrow.

2.Click the graph of the function . Select Translation
from the Construct toolbox, select vector AB and the image of the
graph will appear on the screen.

3.Select the translated image, right-click
and select Properties from the context menu to make Line Style
– Solid 2, and Line Color – Red.

Q3.
What are the coordinates of the
vector of translation that transform the graph of to
the graph of? Make a
conjecture and verify it with the help of the software.

A.
The teacher may help students come
up with a conjecture by pointing to the characteristic points of the graph,
in this case point (0, 0) which is the vertex of the semi-parabola. This
hint should help them determine the coordinates of the vector of translation.
Students will also use the software to help find the answer and verify their
conjecture.

1.Select vector AB, choose Coefficients
from the Constrain toolbox. The coefficients of
the vector will appear in the edit box. Accept the default values by pressing
Enter.

2.By moving the endpoint of the
vector B students can overlap the translated image with the graph of the
function and
observe the values of u0, v0, and a
in the Variables toolbox to answer the question and to verify their
conjecture.

The
expected answer: u0 = a, v0 = 0.

3.In order to verify this,
students can double click the coefficients and substitute these values. The
translated image will overlap with the graph of the function . Drag
point B to confirm this.

Q4. What vector will translate the graph of the function
back to the
graph of the function ? Make a
conjecture and verify with the software.

A. The vector of translation is (-a, 0).

Q5.
Generalize these results for an
arbitrary function f(x) and confirm your conclusion with the
help of the software.

A. Students should conclude that the graph of function
y = f(x – a) is derived from the graph of
function y = f(x) by translation along the y-axis by a
vector (a, 0). They should also conclude that the graph of function y
= f(x) is derived from the graph of function y = f(x
– a) by translation along the x-axis by a vector (-a, 0). They
can verify this by editing the equation of the function:

1.Double click the equation of the
function and change it to: f(x). The graph of the arbitrary function
will appear on the screen

2.Click the image of the
translated function and choose Implicit Equation from the Calculate
toolbox. The equation y = f(x - a) will appear on the screen.

Similarly,
students can verify the inverse translation with the help of the software

In this part of the problem
students will investigate the commutative property of translations given by
vectors (0, b) and (a, 0) along the coordinate axes of the
graph of an arbitrary function y = f(x).

Here
are the steps of construction and questions.

1.Open a new file. If axes do not
appear in the blank document, create them by clicking the Toggle grid and axes
icon on the top toolbar.

Q1.
Make a conjecture about relative
position of the curves transformed by two methods 1) translate the graph of
the function y = f(x) by vector AB and then translate
the resulting graph by the vector CD, or 2) translate the graph of the
function y = f(x) by vector CD and then translate the resulting
graph by the vector AB. Verify your conjecture with the help of the software.

A.
Students will formulate their conjecture
prior to working on the computer, then complete constructions on the computer
and determine coincidence of the graphs, confirming or rejecting their
conjectures. Here are the steps of the constructions.

At
this point, it’s useful to adjust the default Curve setting to the
thinnest black line so that you can see when two curves are the same. From
the drop-down Edit menu at the top of the screen select Settings and
the Geometry tab. In the Curve box check that the Line
Color is Black and the Line Style is Solid 1.

Method
1:

1.Click the graph of the function y
= f(x), select Translation from the Construct
toolbox, then click vector AB. Click the resulting image, right-click and
select Properties from the context menu to change the line style to
Dot.

2.Click the translated image,
select Translation from the Construct toolbox. Then click the
vector CD. Click the translated image, right-click and select Properties
from the context menu and choose Line Style – Solid 4, Line Color
– Light Blue (or any pastel shade).

Method
2:

3.Click the graph of the function y
= f(x) and select Translation from the Construct
toolbox, then click vector CD. Click the resulting image, right-click and
select Properties from the context menu to change the line style to
Dot.

4.With the graph still selected, choose
Translation from the Construct toolbox. Then click vector AB.

Observe
the image constructed by method 2 coincides with the image constructed by
method 1.

Students
should drag point B up and down and point D left and right to verify that
this conjecture works for vectors of various lengths and directions.

Q2. What is the equation of this curve?

A.
It is expected that students will
apply prior knowledge from parts 1 and 2, and will be able to write in either
order:

, or

.
Students then can check their findings using the software – click the curve
and select Implicit Equation from the Calculate toolbox.

Q3.
Is it possible to derive a graph of
the function y = f(x – a) + b from the
graph of the function y = f(x) in one step?

A. It is expected that students will conjecture that
a translation by a vector (a, b) will transform one graph into
another in one step. Students can check that using the software. In order to
do that students will need to add two vectors of translation and then
translate the original graph of function by the resultant vector to verify
that it will coincide with the function y = f(x – a)
+ b

a.Can graph of one of the functions be derived from
the graph of the other function by translations?

b.If yes, by how many ways?

c.If yes, write the vector of translation for the graph
of f to be transformed to the graph of g and visa versa.

d.Confirm your results with the help of the software

A.

The
graph of one of the functions can be derived from the other by
translation along the x-axis

There
are two translations, one from graph of f to g, and
another one from g to f, if a + b¹ 0. If a + b = 0 and thus f(x)
= g(x) then there is only one translation.

The
vectors of translation are (±|a + b|,
0)

Here
are the steps of construction:

1.Click Function from the Draw toolbox
and type sqrt(X-a) for the function f(x). Click OK.
Select the graph of the function, right-click and select Properties from the
context menu, select Line Style – Solid 2, Line Color – Blue.

2.Click Function from the Draw toolbox
and type sqrt(X+b) for the function g(x). Click OK.
Select graph of the function, right-click and select Properties from
the context menu and select Line Style – Solid 2, Line Color –
Red.

3.Click Vector from the Draw toolbox and
construct a vector AB anywhere on the positive y coordinate plane. (The
translation vector needs to reside in the valid domain of the function;
otherwise Geometry Expressions will map the vector’s endpoint to an
indeterminate location.) Click the select arrow. Select the vector,
choose Coefficients from the Constrain toolbox and enter
coefficients |a + b|, 0 for the vector.

4.Select point A and the graph of
. Select Incident from the Constrain toolbox. Point A will lie
on the graph of the function. Point B will automatically be placed on the
graph of .
Students can drag vector AB along the curve to see that the endpoints of the
vector lie on both curves for all points on both graphs.

Likewise,
construct the opposite vector, (-|a+b|, 0) and attach it to function f(x).

Q2. Given two functions, and ,

a.Can the graph of one of the
functions be derived from the graph of the other function by translations?

b.If yes, by how many ways?

c.If yes, write the vector of
translation for the graph of f to be transformed to the graph of g
and visa versa.

d.Confirm your results with the
help of the software

A.

The
graph of one function can be derived from the other one by translation
along the y-axis

There
are two translations, one from graph of f to g, and
another one from g to f, if a¹b If a = b and thus f(x)
= g(x) then there is only one translation.

The
vectors of translation are

Here
are the steps of construction:

Click
Function from the Draw toolbox and type ln(a*X) for the
function f(x). Click OK. Select the graph of f(x),
right-click and select Properties from the context menu, select Line
Style – Solid 2, Line Color – Blue.

Click
Function from the Draw toolbox and type ln(b*X) for the
function g(x). Click OK. Note: If the
default values of a and b are set to the same value,
(check the Variables toolbox), the two functions will be
superimposed. Select one of the variables in the Variables list
and change its value in the edit window below to separate the curves. Select
the graph of g(x),right-click and select Properties
from the context menu, select Line Style – Solid 2, Line Color
– Red.

Click
Vector from the Draw toolbox and construct a vector AB
anywhere on the positive x coordinate plane. (The translation vector
needs to reside in the valid domain of the function; otherwise Geometry
Expressions will map the vector’s endpoint to an indeterminate
location.) Click the selection arrow. Select the vector, choose Coefficients
from the Constrain toolbox and enter for
the vector.

Select
point A and the graph of g(x). Select Incident from
the Constrain toolbox. Point A will lie on the graph of g(x).
Point B will automatically be placed on the graph of f(x).
Students can drag vector AB along the curve to see that endpoints of the
vector lie on both curves for all points on both graphs.

Likewise,
construct the opposite vector, and attach it
to function f(x).

Q3.Given two functions, and ,

a.Can the graph of one of the functions be derived
from the graph of the other function by translations?

b.If yes, by how many ways?

c.Can you specify a translation vector for the graph
of f to be transformed to the graph of g and visa versa?

d.Confirm your results with the help of the software

A.

Yes
if a = c.

There
are an infinite number of vectors that connect any two points on the two
lines.

Students
can specify any vector with endpoints on these two lines or give the
general form of the vector between these two lines

Here
are the steps of construction:

1.Click Function from the Draw toolbox
and type a*X+b for the function f(x). Click OK. Select the
graph of the function, right-click and select Properties from the context
menu, select Line Style – Solid 2, Line Color – Blue.

Comment: if the student
already determined that lines should be parallel, he/she should type the line
with the same slope. If not, the student will start with the general
equations of the lines and will arrive at the conclusion that their slopes
are the same and edit the equation of the function to fit this conclusion.

2.Click Function from the Draw toolbox
and type a*X+d for the function g(x). Click OK. Select the
graph of the function, right-click and select Properties from the context
menu, select Line Style – Solid 2, Line Color – Red.

3.Click Vector in the Draw toolbox and
construct a vector AB anywhere on the plane. Click the selection arrow.

4.Select point A and the graph of f(x).
Select Incident from the Constrain toolbox. The point A will appear
on the graph of f(x).

5.Select graph of f(x), choose Translation from
the Construct toolbox and click vector AB. The translated image of
graph of f(x) will appear on the screen.

6.Move point B around to overlap the translation with
the graph of function g to see various ways that the graph of f
can be translated to graph of g.

Q1.Given two functions, and. What is
value of parameter a and the translation vector needed to equate these
two functions. Investigate this problem with the help of the software

A. Here are the steps of
construction:

1.Click Function from the Draw toolbox
and type 1/X for the function f(x). Click OK. Select the
graph of f(x), right-click and select Properties from the
context menu, select Line Style – Solid 2, Line Color – Blue.

2.Click Function from the Draw toolbox
and type (X+a)/(X+1) for the function g(x). Click OK.
Select the graph of g(x), right-click and select Properties from the
context menu, select Line Style – Solid 2, Line Color – Red.

Note: The students’ graphs
may look different to the image above. Click a in the Variables list,
then manipulate the slider to change the shape of g(x).

3.Click Vector in the Draw toolbox and
construct a vector AB anywhere on the plane. Click the selection arrow.

4.Select the graph of function g(x)
(red), click Translation in the Construct toolbox, then click the
vector AB. The image of the graph of g(x) will appear on the
screen.

5.Vary parameter a and vector AB until the translated
image coincides with the graph of the function f(x) (blue). In
order to vary parameter a and change the shape of the function g,
click-and-drag the graph. A drag handle, labeled a, appears on the
curve. (Alternatively, you can vary the value of a in the Variablestoolbox as described above.) Vary the direction of the translation by
movinging points A and B.

6.When the desired position is found, students can
observe the value of parameter a in the Variables list. They
can also calculate the Real equation for the function g(x)
by selecting the graph of the function, choosing the Real tab in the Calculate
toolbox and selecting Implicit equation.

7.In order to calculate the translation vector,
students select the vector, choose the Real tab in the Calculate
toolbox and select Coefficients. The numerical coefficients of the
vector will appear on the screen. All answers shown on the screen are
numerical approximations and allow students to make a conjecture about exact
solutions.

Students
now can make a conjecture about the values of parameter a and the translation
vector. They should confirm this through analytical work:

.
Translation exists only when a = 2, so that .
Then, . Thus the
graph of g(x) can be derived from the graph of by
translation by the vector (-1, 1). Thus, the inverse translation that
transforms graph of g(x) into the graph of f(x)
is accomplished by the vector (1, -1). In order to verify that, students can
substitute these values for the parameter a and vector AB into
software.

By analogy with given examples the teacher can point
out the associative property of translation. Students can verify that the
order of translation does not affect the result. This property can be
combined with the commutative property of translation.